An additive basis A of a semigroup T is a subset such that
every element of T, up to a finite set of exceptions, may be written
as a sum of one and the same number h of elements from the basis. The minimal
such number h is called the order of the basis. Thus the squares are
a basis of order 4 of the semigroup of natural integers, and the set of primes
is also a basis of that semigroup (of order at most 4 unconditionally,
3 according to Goldbach's conjecture).
We study bases in a class of infinite abelian semigroups, which we term
translatable semigroups. These include all infinite abelian groups as well
as the semigroup of nonnegative integers. We analyse the "robustness" of bases.
Thus we consider essential subsets of a basis A, that is, sets F
such that A \ F is no longer a basis, and which are minimal.
We show that any basis has only finitely many essential subsets. Further,
we will talk about sets whose removal does not destroy the basicity but
increases the order.
Joint work with Benjamin Girard and Thai Hoang Lê.